Editing Directional statistics (section)

== Distribution of the mean ==
Given a set of ''N'' measurements <math>z_n=e^{i\theta_n}</math> the mean value of ''z'' is defined as:

:<math>
\overline{z}=\frac{1}{N}\sum_{n=1}^N z_n
</math>

which may be expressed as

:<math>
\overline{z} = \overline{C}+i\overline{S}
</math>

where

:<math>
\overline{C} = \frac{1}{N}\sum_{n=1}^N \cos(\theta_n) \text{ and } \overline{S} = \frac{1}{N}\sum_{n=1}^N \sin(\theta_n)
</math>

or, alternatively as:

:<math>
\overline{z} = \overline{R}e^{i\overline{\theta}}
</math>

where

:<math>
\overline{R} = \sqrt{{\overline{C}}^2+{\overline{S}}^2} \text{ and } \overline{\theta} = \arctan (\overline{S} / \overline{C}).
</math>

The distribution of the mean angle (<math>\overline{\theta}</math>) for a circular pdf ''P''(''&theta;'') will be given by:

:<math>
P(\overline{C},\overline{S}) \, d\overline{C} \, d\overline{S} =
P(\overline{R},\overline{\theta}) \, d\overline{R} \, d\overline{\theta} = 
\int_\Gamma \cdots \int_\Gamma \prod_{n=1}^N \left[ P(\theta_n) \, d\theta_n \right]
</math>

where <math>\Gamma</math> is over any interval of length <math>2\pi</math> and the integral is subject to the constraint that <math>\overline{S}</math> and <math>\overline{C}</math> are constant, or, alternatively, that <math>\overline{R}</math> and <math>\overline{\theta}</math> are constant.

The calculation of the distribution of the mean for most circular distributions is not analytically possible, and in order to carry out an analysis of variance, numerical or mathematical approximations are needed.{{sfn|Jammalamadaka|Sengupta|2001}}

The [[central limit theorem]] may be applied to the distribution of the sample means. (main article: [[Central limit theorem for directional statistics]]). It can be shown{{sfn|Jammalamadaka|Sengupta|2001}} that the distribution of <math>[\overline{C},\overline{S}]</math> approaches a [[bivariate normal distribution]] in the limit of large sample size.